Mid-Point Theorem: Statement and Proof

According to the Mid-point Theorem, a line segment drawn from the midpoints of two sides of a triangle is parallel to the third side and equal to half of the third side.

• Geometry is an important branch of mathematics concerned with different shapes and figures.
• Triangles are an essential part of geometry, and the mid-point theorem focuses on their midpoints.
• A triangle is a polygon with three corners and three sides.
• The Mid-Point Theorem is an important theorem in geometry that deals with triangle properties.
• In coordinate geometry, the midpoint theorem states that the midpoint of a line segment is equal to the average of its ends.
• To solve an equation using this theorem, we must know the 'x' and 'y' coordinates.
• The Mid-Point Theorem also has applications in calculus and algebra.

Key Terms: Triangle, Mid-Point Theorem, Converse of Mid-Point Theorem, Geometry, Similar triangle, Midpoint, Parallel lines, Bisector, Alternate interior angles

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Mid-Point Theorem Statement

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According to the mid-point theorem,

“The line segment connecting the midpoints of any two sides of a triangle is parallel to the third side of the triangle and is also half of the length of the third side.”

Consider an arbitrary triangle ΔABC.

• Let D and E represent the midpoints of AB and AC, respectively.
• Draw a straight line from D to E.
• The midpoint theorem states that DE will be parallel to BC and equal to exactly half of BC.

Look at the image below to understand the triangle midpoint theorem.

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Mid-Point Theorem Proof

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Consider the triangle ABC. Let D and E be the midpoints of the sides AB and AC respectively.

To Prove 

DE ∥ BC and DE = 1/2(BC)

Construction

Extend the line segment that connects points D and E to F so that DE = EF and joins CF.

Proof

In ∆AED and ∆CEF 

• DE = EF (construction)
• ∠1 = ∠2 (vertically opposite angles)
• AE = CE (E is the mid-point)
• △AED is similar to △CEF i.e. △AED ≅ △CEF (By Side-angle-side condition)

By corresponding parts of the congruent triangle (c.p.c.t), we have

∠3 =∠4

But these are alternate interior angles.

Therefore, AB ∥ CF

⇒ AD = CF  (c.p.c.t)

But AD = DB (D is the mid-point)

Therefore, BD = CF

In BCFD, we have

BD ∥ CF (as AB ∥ CF)

⇒ BD = CF

BCFD is a parallelogram as one pair of opposite sides is parallel and equal.

Therefore, 

• DF∥ BC (opposite sides of the parallelogram)
• DF = BC (opposite sides of the parallelogram)

As DF ∥ BC, DE ∥ BC and DF = BC

But DE = EF

Therefore, DF = 2(DE)

⇒ 2(DE) = BC

⇒ DE = 1/2(BC)

Therefore it proved that the line joining the midpoints of two sides of the triangle is parallel to and half the length of the third side.

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Mid-Point Theorem Formula

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In Coordinate Geometry, the midpoint theorem describes the midpoint of a line segment. It defines the coordinate points of the midpoint of the line segment and can be found by taking the average of the coordinates of the given endpoints. The midpoint formula is used to find the midpoint between two given points.

The midpoint formula for the coordinates of two given endpoints P₁(x₁, y₁) and P₂(x₂, y₂) is given by

\(Midpoint = \left[ \frac {(x_1 + x_2)}{2}, \frac {(y_1 + y_2)}{2} \right]\)

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The Converse of Mid-Point Theorem

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The Converse of the Midpoint theorem states that

"If a line is drawn across the midpoint of one side of a triangle and parallel to the second side, it bisects the third side".

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Solved Examples

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Ques. In the figure given below L, M, and N are mid-points of side PQ, QR, and PR respectively of triangle PQR. If PQ = 7 cm, QR = 8 cm and PR = 5 cm. Find the perimeter of the triangle formed by joining L, M, and N.

Ans. Given that, L and N are mid-points, therefore by the mid-point theorem, we have

• LN ∥ QR
• LN = 1/2 × (QR)

Hence, LN = 1/2 × 8 = 4 cm

Similarly, LM = 1/2 × (PR) = 1/2 × (5) = 2.5 cm

Also, MN = 1/2 × (PQ) = 1/2 × (7) = 3.5 cm

Therefore,

Perimeter of △LMN = LM + MN + LN

Perimeter of △LMN = 2.5 + 3.5 + 4 = 10 cm

Therefore, the perimeter of △LMN is 10 cm.

Ques. In triangle ABC, the midpoints of BC, CA, and AB are D, E, and F, respectively. Find the value of EF, if the value of BC = 18 cm.

Ans. Given: BC = 18 cm

If F is the midpoint of AB and E is the midpoint of AC, then using the midpoint theorem, we get

EF = 1/2 (BC)

On substituting the value of BC, we get

EF = (1/2) × 18

EF = 9 cm

Therefore, the value of EF = 9 cm.

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Things to Remember

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• Geometry is an important branch of mathematics concerned with different shapes and figures.
• According to the mid-point theorem, “The line segment connecting the midpoints of any two sides of a triangle is parallel to the third side of the triangle and is also half of the length of the third side.”
• The mid-Point Formula is given by, \(Midpoint = \left[ \frac {(x_1 + x_2)}{2}, \frac {(y_1 + y_2)}{2} \right]\)
• According to the converse of the mid-point theorem, If a line is drawn across the midpoint of one side of a triangle and parallel to the second side, it bisects the third side.
• The Mid-Point Theorem also has applications in calculus and algebra.

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